Decomposition of Time Series

Generally, we can categorize patterns in time series data into the following two categories:

• Structural patterns - also known as the series components. There are three main components:
  • trend - the series growth overtime
  • cycle - repeated events over time that are not equally spaced
  • seasonal - repeated events over time that are equally spaced
• Non-structural patterns - also known as the irregular components, which is basically any pattern on the series that is not represented by the structural patterns. An example of such a pattern could be repeated events that are not seasonal (e.g., Easter or the Chinese New Year are both occurred every year, but the date is changing from year to year)

Those patterns are either exists or not (e.g., a series may have a trend or not). We can express those components using the following notation for additive structure:

\[Y_t=T_t + C_t + S_t + I_t, \]

And for multiplicative structure:

\[Y_t=T_t \times C_t \times S_t \times I_t \]

For simplicity reasons, we will joined the cycle component into the trend, and rewrite the series components notation for additive structure:

\[Y_t=T_t + S_t + I_t, \] And for multiplicative structure:

\[Y_t=T_t \times S_t \times I_t \]

In this section, we will focus on decomposition methods of time series to its components - the trend, seasonal, and irregular.

Classical Decomposition

The classical decomposition (or by its full name - classical seasonal decomposition by moving average) is one of the most commom estimation method of the series components. R provides a built-in method for decompose time series with the decompose function from the stats package:

data("AirPassengers")

d <- decompose(AirPassengers)

str(d)

## List of 6
##  $ x       : Time-Series [1:144] from 1949 to 1961: 112 118 132 129 121 135 148 148 136 119 ...
##  $ seasonal: Time-Series [1:144] from 1949 to 1961: -24.75 -36.19 -2.24 -8.04 -4.51 ...
##  $ trend   : Time-Series [1:144] from 1949 to 1961: NA NA NA NA NA ...
##  $ random  : Time-Series [1:144] from 1949 to 1961: NA NA NA NA NA ...
##  $ figure  : num [1:12] -24.75 -36.19 -2.24 -8.04 -4.51 ...
##  $ type    : chr "additive"
##  - attr(*, "class")= chr "decomposed.ts"

plot(d)

d_m <- decompose(AirPassengers, type = "multiplicative")

plot(d_m)

library(TSstudio)

ts_decompose(AirPassengers, type = "both")

library(tsibble)
library(feasts)
library(fabletools)

ap_tsibble <- as_tsibble(AirPassengers)

decompose_md <- ap_tsibble %>% 
  model(classical_decomposition(value, type = "multiplicative"))

decompose_md %>% 
  components() %>% 
  head()

## # A dable:                 6 x 7 [1M]
## # Key:                     .model [1]
## # Classical Decomposition: value = trend * seasonal * random
##   .model                                                         index value trend seasonal random season_adjust
##   <chr>                                                          <mth> <dbl> <dbl>    <dbl>  <dbl>         <dbl>
## 1 "classical_decomposition(value, type = \"multiplicative\")" 1949 Jan   112    NA    0.910     NA          123.
## 2 "classical_decomposition(value, type = \"multiplicative\")" 1949 Feb   118    NA    0.884     NA          134.
## 3 "classical_decomposition(value, type = \"multiplicative\")" 1949 Mar   132    NA    1.01      NA          131.
## 4 "classical_decomposition(value, type = \"multiplicative\")" 1949 Apr   129    NA    0.976     NA          132.
## 5 "classical_decomposition(value, type = \"multiplicative\")" 1949 May   121    NA    0.981     NA          123.
## 6 "classical_decomposition(value, type = \"multiplicative\")" 1949 Jun   135    NA    1.11      NA          121.

decompose_md %>% 
  components() %>%
  autoplot()

d_tsibble <- decompose_md$`classical_decomposition(value, type = "multiplicative")`[[1]]$fit$decomposition

STL Decomposition

WIP